Solving For Zeros Of Quadratic Function F(x) = X + 5 - 2x² A Step-by-Step Guide
In the realm of mathematics, quadratic functions hold a significant position due to their widespread applications in various fields, ranging from physics and engineering to economics and computer science. A quadratic function is a polynomial function of degree two, typically expressed in the standard form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The zeros of a quadratic function, also known as the roots or x-intercepts, are the values of x for which the function f(x) equals zero. Finding these zeros is a fundamental task in algebra, as they provide crucial information about the function's behavior and its graph, a parabola. In this comprehensive guide, we will delve into the process of solving for the zeros of a quadratic function, focusing on the specific example f(x) = x + 5 - 2x². We will meticulously walk through each step, providing clear explanations and illustrative examples to ensure a thorough understanding of the underlying concepts. Our journey will begin with transforming the given quadratic function into its standard form, a crucial step for applying various solution methods. Next, we will set the function equal to zero, transforming the problem into a quadratic equation. Then, we will identify the coefficients a, b, and c, which are essential for employing methods like the quadratic formula or factoring. Finally, we will apply these methods to determine the zeros of the function, interpreting the results in the context of the function's graph and its real-world applications. By the end of this exploration, you will have a solid grasp of how to solve for the zeros of quadratic functions, empowering you to tackle a wide range of mathematical problems and real-world scenarios.
The initial step in finding the zeros of a quadratic function is to express it in its standard form. The standard form of a quadratic function is given by f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. This form is crucial because it allows us to easily identify the coefficients a, b, and c, which are necessary for applying various methods to solve for the zeros, such as the quadratic formula or factoring. The given quadratic function is f(x) = x + 5 - 2x². To convert this function into standard form, we need to rearrange the terms so that the term with x² comes first, followed by the term with x, and finally the constant term. This rearrangement involves applying the commutative property of addition, which states that the order of terms in an addition does not affect the sum. By rearranging the terms, we get f(x) = -2x² + x + 5. Now, the function is in standard form, and we can clearly identify the coefficients. Here, the coefficient of the x² term, a, is -2. The coefficient of the x term, b, is 1 (since x is the same as 1x). And the constant term, c, is 5. Identifying these coefficients is a critical step, as they are the key ingredients for the next stages of solving for the zeros. The standard form not only makes it easier to apply solution methods but also provides insights into the shape and position of the parabola, which is the graph of the quadratic function. The coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0), and the vertex of the parabola can be found using the coefficients a and b. Thus, putting the quadratic function in standard form is a fundamental step that sets the stage for further analysis and problem-solving.
After expressing the quadratic function in standard form, the next pivotal step in solving for its zeros is to set the function equal to zero. This transformation converts the quadratic function into a quadratic equation, which is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation, also known as the roots, are precisely the zeros of the original function. The zeros of a function are the values of x for which the function's output, f(x), is zero. Graphically, these zeros correspond to the points where the parabola intersects the x-axis. These points are also referred to as the x-intercepts of the graph. For the given quadratic function, f(x) = -2x² + x + 5, setting it equal to zero yields the quadratic equation -2x² + x + 5 = 0. This equation represents the core of our problem: finding the values of x that satisfy this equation will give us the zeros of the function. Solving a quadratic equation can be accomplished through various methods, including factoring, completing the square, and the quadratic formula. The choice of method often depends on the specific form of the equation and the ease with which it can be manipulated. For instance, if the quadratic expression can be easily factored, then factoring is often the most efficient method. However, for more complex equations, the quadratic formula provides a reliable and universally applicable solution. The act of setting the function equal to zero is not merely a mathematical manipulation; it is a conceptual shift that allows us to reframe the problem from finding the inputs that make the function zero to finding the solutions of an equation. This shift is crucial because it opens up a wide array of algebraic techniques and tools that can be used to solve for the unknown variable, x. Therefore, setting the function equal to zero is a fundamental step in the process of finding the zeros of a quadratic function.
With the quadratic equation established, the next crucial step involves identifying the values of the coefficients a, b, and c. These coefficients are the numerical constants that define the quadratic equation in its standard form, ax² + bx + c = 0. Accurate identification of these values is paramount because they are the key inputs for various methods used to solve quadratic equations, such as the quadratic formula and factoring techniques. In our specific example, the quadratic equation is -2x² + x + 5 = 0. By comparing this equation to the standard form, we can readily determine the values of a, b, and c. The coefficient a is the numerical value multiplying the x² term. In this case, a = -2. The coefficient b is the numerical value multiplying the x term. Here, b = 1 (since x is equivalent to 1x). Finally, the coefficient c is the constant term, which is the term without any x variable. In our equation, c = 5. These values, a = -2, b = 1, and c = 5, are now ready to be used in the next stage of the solution process, which typically involves applying the quadratic formula or attempting to factor the quadratic expression. The significance of correctly identifying a, b, and c cannot be overstated. An error in these values will inevitably lead to incorrect solutions for the quadratic equation. Therefore, it is essential to pay close attention to the signs and magnitudes of the coefficients when extracting them from the equation. Furthermore, understanding the role of each coefficient provides deeper insights into the behavior of the quadratic function. The coefficient a determines the parabola's direction (upward if a > 0, downward if a < 0) and its vertical stretch or compression. The coefficient b influences the position of the parabola's axis of symmetry, and the coefficient c represents the y-intercept of the parabola. Thus, identifying a, b, and c is not just a mechanical step; it is a crucial aspect of understanding the quadratic function and its graphical representation.
The quadratic formula is a powerful and universally applicable tool for solving quadratic equations of the form ax² + bx + c = 0. It provides a direct method for finding the roots of the equation, regardless of whether the equation can be easily factored or not. The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation. The "±" symbol indicates that there are two possible solutions, one obtained by adding the square root term and the other by subtracting it. To apply the quadratic formula to our equation, -2x² + x + 5 = 0, we first recall the values of the coefficients that we identified earlier: a = -2, b = 1, and c = 5. Now, we substitute these values into the formula:
x = (-1 ± √(1² - 4(-2)(5))) / 2(-2)
Next, we simplify the expression step by step. First, we calculate the term inside the square root:
1² - 4(-2)(5) = 1 + 40 = 41
So, the equation becomes:
x = (-1 ± √41) / -4
Now, we have two possible solutions:
x₁ = (-1 + √41) / -4 x₂ = (-1 - √41) / -4
These are the zeros of the quadratic function. We can approximate these values using a calculator:
x₁ ≈ (-1 + 6.403) / -4 ≈ -1.351 x₂ ≈ (-1 - 6.403) / -4 ≈ 1.851
Thus, the zeros of the quadratic function f(x) = -2x² + x + 5 are approximately -1.351 and 1.851. The quadratic formula is a cornerstone of algebra, providing a reliable method for solving any quadratic equation. Its power lies in its generality; it can handle equations with real or complex roots, and it does not depend on the ability to factor the quadratic expression. The term inside the square root, b² - 4ac, is called the discriminant. The discriminant provides valuable information about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (a repeated root). And if it is negative, the equation has two complex roots. In our case, the discriminant is 41, which is positive, confirming that we have two distinct real roots.
The zeros of the quadratic function, which we found to be approximately -1.351 and 1.851, have significant meaning in the context of the function's graph and its real-world applications. The zeros represent the x-intercepts of the parabola, which is the graphical representation of the quadratic function. These are the points where the parabola intersects the x-axis, meaning that at these points, the value of the function f(x) is zero. In our example, the parabola intersects the x-axis at approximately x = -1.351 and x = 1.851. These points provide a visual understanding of the function's behavior. The zeros also divide the x-axis into intervals where the function's values are either positive or negative. Between the zeros, the parabola lies either above or below the x-axis, indicating whether the function's values are positive or negative in that interval. Outside the zeros, the parabola extends in the same direction, either above or below the x-axis, indicating the sign of the function's values in those intervals. The zeros are also crucial in determining the vertex of the parabola, which is the point where the function reaches its maximum or minimum value. The x-coordinate of the vertex is located at the midpoint of the zeros. In our case, the x-coordinate of the vertex is approximately (-1.351 + 1.851) / 2 = 0.25. By substituting this value back into the function, we can find the y-coordinate of the vertex, which represents the maximum or minimum value of the function. Beyond their graphical interpretation, the zeros of a quadratic function have practical applications in various fields. For instance, in physics, the zeros might represent the times at which a projectile hits the ground. In economics, they could represent the break-even points for a business. In engineering, they might represent the points of stability in a system. The ability to find and interpret the zeros of a quadratic function is a fundamental skill in mathematics, providing insights into the function's behavior and its relevance to real-world scenarios. The zeros are not just numbers; they are key pieces of information that unlock a deeper understanding of the quadratic function and its applications.
In this comprehensive exploration, we have successfully navigated the process of solving for the zeros of the quadratic function f(x) = x + 5 - 2x². We began by transforming the function into its standard form, f(x) = -2x² + x + 5, a crucial step for identifying the coefficients a, b, and c. We then set the function equal to zero, converting it into a quadratic equation, -2x² + x + 5 = 0. Next, we meticulously identified the values of the coefficients: a = -2, b = 1, and c = 5. With these values in hand, we applied the quadratic formula, a powerful tool that provides a direct method for finding the roots of any quadratic equation. Through careful substitution and simplification, we obtained the zeros of the function, which are approximately -1.351 and 1.851. Finally, we delved into the interpretation of these zeros, understanding their significance as the x-intercepts of the parabola, the graphical representation of the quadratic function. We discussed how the zeros divide the x-axis into intervals, indicating where the function's values are positive or negative, and how they relate to the vertex of the parabola. We also highlighted the practical applications of zeros in various fields, from physics and economics to engineering. This journey through solving for the zeros of a quadratic function underscores the importance of a systematic approach, attention to detail, and a deep understanding of the underlying concepts. The quadratic formula, in particular, stands out as a versatile tool that empowers us to tackle a wide range of quadratic equations. The ability to find and interpret the zeros of a quadratic function is a fundamental skill in mathematics, providing insights into the function's behavior and its relevance to real-world scenarios. As you continue your mathematical journey, remember that the zeros are not just numbers; they are key pieces of information that unlock a deeper understanding of the quadratic function and its applications. By mastering the techniques presented here, you will be well-equipped to solve a variety of mathematical problems and apply these concepts to real-world situations.
